An edge-disjoint weak \(T\) -linkage \(\mathcal {W}\)  can be considered as a flow function
\(\phi : E \rightarrow \Sigma ^*\) .
Let \(W_i\)  be the path of \(\mathcal {W}\)  connecting \(s_i\)  and \(t_i\)  for \(i\in [k]\) .
For each edge \(e\in E(G)\) , let \(\phi (e)=i\)  if it is used in the path connecting \(s_i\)  to \(t_i\) , and
let \(\phi (e)=\epsilon \)  if it is not used by any path of \(\mathcal {W}\) .
We say \(\phi \)  is the flow function representing \(\mathcal {W}\) .
But it is possible that a flow function \(\phi \)  represents more than two weak \(T\) -linkages of \(G\) .
For illustration, refer to Figure 2 of [1]}.
Schrijver [2]} showed that a flow function \(\phi \)  represents a \(T\) -linkage of \(G\) 
if and only if \(\phi (\pi )\in \Sigma \cup \Sigma ^{-1}\)  for every face-edge path consisting of faces and edges sharing a common vertex.
Here, a face-edge path in \(G\)  is an alternating sequence of faces and edges
such that every face is incident (in \(G\) ) to the edges neighboring in the sequence.
